Coherent phase equilibrium in alloys with congruent points
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I.
INTRODUCTION
THE effects of elastic
stress on the equilibrium state of a two-phase coherent alloy have become a subject of great interest recently. Beginning with the work of Williams [~,2] and continuing through the recent publications of Liu and Agren [31 and Johnson and Miiller, [4J it has become clear that elastic stress can greatly alter the nature of two-phase equilibrium from that found in the absence of stress. For example, elastic stress can preclude regions of two-phase coexistence which appear on the stress-free phase diagram [1-3,51 while also inducing metastable two-phase coexistence. [4,6,7.9] Elastic stress can also cause the compositions of the phases to be a function of the alloy composition in a binary alloy and result in discontinuous jumps in the volume fractions of the phases with a continuous change in temperature or alloy composition. While all of this past work has shown that elastic stress can alter radically the nature of phase equilibrium, to date, it has been difficult to make general statements concerning the effects of elastic stress on phase equilibrium. In part, this is because of the strong dependence of the elastic state of the two-phase system on, for example, the morphology second-phase particles, the spatial distribution of the particles, and the degree of elastic constant anisotropy. However, it is also related to the fact that it has been necessary to either make assumptions on the detailed stress-free chemical thermodynamics of the system [3,5-s~ or assume that the changes in phase compositions are small and linearize the equilibrium equations about the stress-free state. [9J
M.J. PFEIFER, Graduate Student, and P.W. VOORHEES, Associate Professor, are with the Department of Materials Science and Engineering, McCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL 60208. Manuscript submitted September 26, 1990. METALLURGICAL TRANSACTIONS A
Here, we present a description of coherent phase equilibria which is valid regardless of the detailed chemical thermodynamics of the alloy, but still under specific assumptions on the stress state of the two-phase system. We have chosen to examine an alloy system which displays a congruent point in the absence of stress. The presence of a congruent point allows the chemical thermodynamics to be described using Taylor series expansions of the stress-free free energies of each of the phases about the congruent temperature and composition. As the free energies of the phases are necessarily analytic at the congruent point, the Taylor expansions yield quite realistic results. Thus, in some sense, this approach is similar to a Landau expansion for a system displaying a critical consulate point, and as a result, the description of coherent phase equilibrium will have much of the well-known advantages and disadvantages of Landau expansions. Specifically, the description is valid regardless of the detailed chemical thermodynamics of the system, captures some of the fundamental physics underlying phase equilib
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